Page:The principle of relativity (1920).djvu/158

 impulse-vector, and m-times the acceleration-vector at P as the force-vector of motion, at P. According to these definitions, the following law tells us how the motion of a point-mass takes place under any moving force-vector : The force-vector of motion is equal to the moving force-vector.

This enunciation comprises four equations for the components in the four directions, of which the fourth can be deduced from the first three, because both of the above-mentioned vectors are perpendicular to the velocity-vector. From the definition of T, we see that the fourth simply expresses the "Energy-law." Accordingly c^2-''times the component of the impulse-vector in the direction of the t-axis is to be defined as the kinetic-energy'' of the point-*mass. The expression for this is

mc^2 dt/dτ = mc^2 / [sqrt](1 - v^2/c^2)

i.e., if we deduct from this the additive constant mc^2, we obtain the expression 1/2 mv^2 of Newtonian-mechanics upto magnitudes of the order of 1/c^2. Hence it appears that ''the energy depends upon the system of reference''. But since the t-axis can be laid in the direction of any time-like axis, therefore the energy-law comprises, for any possible system of reference, the whole system of equations of motion. This fact retains its significance even in the limiting case c = [infinity], for the axiomatic construction of Newtonian mechanics, as has already been pointed out by T. R. Schütz. Nachr. 1897, p. 110.]